The Fourier Transform (FT) is one of the most important signal processing techniques in use today. In particular, it finds a number of uses in medical imaging, including reconstruction of MRI images and Fourier reconstruction of CT images. In MRI applications, the FT is used to convert the acquired data into an image. The quality and accuracy of the image is of utmost importance in medical examinations.
The Fast Fourier Transform (FFT), is an efficient implementation of the FT, which can only be utilized on data that is uniformly sampled in a transform domain. In addition to the FFT, there are other signal and image processing techniques that require the input data to be sampled to a specific grid, for example, backprojection reconstruction in CT or MRI imaging and diffraction tomography.
In many real-world situations, the data is not uniformly sampled. In spiral MRI, for example, the non-uniform sampling is due to variations in magnetic gradients and timing circuits. Typically, allowing non-uniform sampling can significantly shorten MRI data acquisition time.
In MRI imaging, data is acquired into a signal space called a k-space, which is the Fourier transform space of the image. An image is usually reconstructed from the k-space by applying an FFT to the data in the k-space. A major difference between MRI methods is the order in which data is acquired into the k-space. For example, in spiral MRI, data is acquired along a spiral trajectory in a two dimensional k-space, while in spin echo MRI, data is acquired along individual rows in the k-space. In 3D MRI, the k-space is three dimensional.
The commonly used solution to non-uniformly spaced data points is to interpolate the data points onto a uniformly spaced grid. One method of interpolation (referred to herein as the GRD method) was originally devised for radio astronomy by W. N. Brouw in “Aperture Synthesis”, B. Adler, S. Fernbach and M. Rotenberg, editors, Methods in Computational Physics, Vol. 14, pp. 131–175, Academic press, New York, 1975. This method was introduced into medical imaging by O'Sullivan in “A Fast Sinc Function Gridding Algorithm for Fourier Inversion”, IEEE Trans. Med. Imaging, MI4:200–207, 1985 and by Jackson et. al in “Selection of A Convolution Function for Fourier inversion using Gridding”, in IEEE Trans. Med. Imaging, MI-10:473–478, 1991, and further elaborated by Meyer et. al in “Fast Spiral Coronary Artery Imaging”, Magn. Reson. Med., 28:202–213, 1992, the disclosures of which are incorporated herein by reference.
In a conventional gridding method, resampling of data (in k-space) is applied as follows:
(a) Pre-compensate the sampled data with the inverse of the sampled data density, to compensate for the varying density of sampling in k-space.
(b) Convolute the data with a Kaiser-Bessel window function.
(c) Re-sample onto a uniformly spaced Cartesian grid.
(d) Perform an FFT on the redistributed set of data points to get an image.
(e) Post-compensate the transformed data to remove apodization of the convolution kernel by dividing the image data by the transform of the Kaiser-Bessel window function.
A Kaiser-Bessel convolution is used, rather than a sinc convolution, to reduce the computational complexity. The preferred convolution kernel is zero outside a certain window size, so that each resampled data point will be interpolated only from a small number of data points, in its vicinity, rather than using all or most of the data points in the set, as would be required in a sinc based interpolation.
The step of post-compensation is required to correct for a so-called roll-off effect induced by the transform of the convolution window. Even after the post compensation, there is generally a degradation of image quality towards the edges of the generated image. Two types of effects are generally visible: “cupping” and “wings”. Cupping is where the intensity profile is lower (or higher) at the center of the image than at its ends and wings is where there is an overrun of the signal beyond the ends of the image carrying portion of the image. “Density Compensation Functions for Spiral MRI”, by R. Hoge et al., in MRM 38:117–128 (1997), the disclosure of which is incorporated herein by reference, provides in FIG. 5D thereof a graphic example of both degradation effects for a Jackson type gridding algorithm. Such degradation is undesirable in medical images that are used for diagnosis. One solution is to interpolate onto a 2N×2N grid, rather than onto an N×N grid (oversampling). The result is then post-compensated and only the central N×N portion of the post-compensated result is persevered. Most of the artifacts are outside this central portion. However, this technique increases the number of points for the FFT, by a factor of four, which considerably increases the complexity of the computation.
In the performance of real time MRI imaging, e.g. imaging of the heart and imaging fluid dissipation in tissues, the number of computational steps allowed between sequential images should be kept to a minimum. Typically, the pre-compensation, convolution and resampling are performed by multiplying the column-stacked data by a suitable, pre-calculated, matrix of coefficients. In some cases, the pre-compensation is applied separately. The post-compensation requires an element-by-element multiplication by a pre-calculated matrix. The number of required calculations is very important in medical imaging since the size of MRI images can be as large as 1,024×1,024 or more.
An article titled “Comparison of Interpolating Methods for Image Resampling” by J. Parker et al, IEEE trans. on Medical Imaging March 1983, states that the choice of an interpolating function for resampling depends upon the task being performed. When verisimilar images are desired, this article suggests cubic B-spline interpolation. When additional processing of the images is to be performed the article suggests high-resolution cubic spline interpolation.
Another source of image degradation is noise. Substantially every sampled data includes noise, such as, thermal noise from the source of the sampled data and noise due to the apparatus used in acquiring the data. In resampling, the noise in the original data is passed over to the resampled data.
In medical images, certain types of noise are found to be more tolerable to the human observer than others. For example, as mentioned in the above referenced article, noise which is correlated with an image is much more noticeable than noise which is uncorrelated with the image. In viewing medical images, it is commonly desired to receive images which have a clear appearance, i.e. a low level of local noise, even if the received images are less accurate, i.e., have a higher bias.
U.S. Pat. No. 4,982,162 to Zakhor et al. and an article from the same author titled “Optimal sampling and Reconstruction of MRI Signals Resulting from Sinusoidal Gradients”, IEEE transactions on signal processing, September 1991, the disclosures of which are incorporated herein by reference, describe derivation of a one-dimensional least square estimator matrix for generating an image from non-uniform sampled data, based on estimation theory. The estimator requires a matrix inversion which is time consuming. The time required for matrix inversion is a function of the number of sampled data points, which in two dimensional images is on the order of tens of thousands.